1 models with a lagged dependent variable a widely-used solution to the problem of including...

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MODELS WITH A LAGGED DEPENDENT VARIABLE

A widely-used solution to the problem of including dynamics in a model while mitigating the problem of multicollinearity is to employ an autoregressive distributed lag model, often written ADL(p, q).

tttt uYXY 1321 ADL(p, q)

2

The ‘autoregressive’ part of the name refers to the fact that lagged values of the dependent variable are included on the right side as explanatory variables.



ADL = autoregressive distributed lag

'autoregressive' because Yt depends on previous values of Y.

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p is the maximum number of lags of the dependent variable used in this way. q is the maximum lag of the X variable, or variables if there are several.





p = maximum number of lags of the dependent variable

q = maximum number of lags of the X variable(s)

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The ADL model is particularly appealing when the dependent variable exhibits a high degree of dependence because then, as a matter of common sense, its value in one observation is likely to be influenced by its value in the previous one.







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It is econometrically attractive because it can accommodate a broad range of dynamic patterns with relatively few lag terms and parameters. (It is parsimonious, to use the technical term.)







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This is likely to reduce (but obviously, not eliminate) the problem of multicollinearity.







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We will start with the simplest model of all, the ADL(1,0) model where the only lagged variable is the lagged dependent variable.


tttt uYXY 1321 ADL(1, 0)



p = 1 in this specification

q = 0

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Given the continuity of many time series processes, Yt–1, the value of a time series at timet – 1, is often the most important determinant of its value Yt at time t. and it makes sense to include it explicitly in the model as an explanatory variable.


tttt uYXY 1321



p = 1 in this specification

q = 0

ADL(1, 0)

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We will begin by investigating the dynamics implicit in the model graphically. We will suppose, for convenience, that b2 is positive and that X increases with time, and we will neglect the effect of the disturbance term.


b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1

short-run relationship at time t

tttt uYXY 1321

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We shall suppose throughout this section that │b3│ < 1. This is a stability condition for the process. We will discuss the consequences of violations of this condition in Chapter 13. We will in fact assume 0 < b3 < 1 because Yt and Yt–1 are typically positively correlated.


b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1


tttt uYXY 1321

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At time t, Yt is given by the equation at the top. It is represented by the point corresponding to Xt on the lowest of the five lines in the figure. Yt–1 has already been determined at time t, so the term b3Yt–1 is fixed.


b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1


tttt uYXY 1321

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The equation thus may be viewed as giving the short-run relationship between Yt and Xt for period t. (b1 + b3Yt–1) is effectively the intercept and b2, the slope coefficient, gives the short-run effect of X on Y.


b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1


tttt uYXY 1321

13

When we come to time t + 1, Yt+1 is given by the second equation and the effective intercept is now (b1 + b3Yt). Since X is increasing, Y is increasing, so the intercept is larger than that for Yt and the short-run relationship has shifted upwards. The slope is unchanged, b2.


b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1


tttt uYXY 1321

131211 tttt uYXY

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Thus two factors are responsible for the growth of Y over time: the direct effect of the increase in X, and the gradual upward shift of the short-run relationship. The figure shows the outcomes for time t as far as time t + 4.


b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1


tttt uYXY 1321

131211 tttt uYXY

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You can see that the actual relationship between Y and X, traced out by the markers representing the observations, is steeper than the short-run relationship for each time period.


b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1


tttt uYXY 1321

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We will determine the long-run relationship between Y and X by performing a comparative statics analysis described in the previous slideshow.


b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1


tttt uYXY 1321

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Denoting equilibrium Y and X by Y and X, Yt = Yt–1 = Y and Xt = X in equilibrium. Hence, ignoring the transient effect of the disturbance term, the equilibrium relationship is as shown above.

tttt uYXY 1321

b1 + b3Yt+3

Xt Xt+1 Xt+3 Xt+4 X

Y

Xt+2

b1 + b3Yt+2

b1 + b3Yt+1

b1 + b3Yt

b1 + b3Yt–1


YXY 321


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Re-arranging, one has the equilibrium value of Y in terms of the equilibrium value of X. The factor b2 / (1 – b3) gives the effect of a one-unit change in equilibrium X on equilibrium Y. We will describe this as the long-run effect.

XY3

2

3

1

11


tttt uYXY 1321

YXY 321

3

2

1

long-run effect of X on Y

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In the present context, with 0 < b3 < 1, it will be greater than b2 because 1 – b3 will also lie between 0 and 1.


XY3

2

3

1

11

tttt uYXY 1321

YXY 321

3

2

1

long-run effect of X on Y

101 32

3

2

if

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Another way of exploring the dynamics is to look at the implicit relationship between Yt and current and lagged values of X. If the relationship is true for time period t, it is also true for time period t – 1.


tttt uYXY 1321

1231211 tttt uYXY

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We substitute for Yt–1 in the first equation.


tttt uYXY 1321

1231211 tttt uYXY

132

23132231

123121321

1

ttttt

tttttt

uuYXX

uuYXXY

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Continuing to lag and substitute, one obtains the equation shown.


tttt uYXY 1321

1231211 tttt uYXY

132

23132231

123121321

1

ttttt

tttttt

uuYXX

uuYXXY

...

......1

12313

22321322

2331

ttt

tttt

uuu

XXXY

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Hence Yt can be viewed as a linear combination of current and lagged values of X with a lag distribution that consists of geometrically declining weights: b2, b2b3, b2b3

2, ...


tttt uYXY 1321

1231211 tttt uYXY

132

23132231

123121321

1

ttttt

tttttt

uuYXX

uuYXXY

...

......1

12313

22321322

2331

ttt

tttt

uuu

XXXY

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We have thus found a way of allowing lagged values of X to influence Y without introducing them into the model explicitly and giving rise to multicollinearity.


tttt uYXY 1321

1231211 tttt uYXY

132

23132231

123121321

1

ttttt

tttttt

uuYXX

uuYXXY

...

......1

12313

22321322

2331

ttt

tttt

uuu

XXXY

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One should, however, note that this particular pattern of weights, known as a Koyck lag distribution, embodies the assumption that more recent values of X have more influence than older ones, and that the rate of decline in the weights is constant.


tttt uYXY 1321

1231211 tttt uYXY

132

23132231

123121321

1

ttttt

tttttt

uuYXX

uuYXXY

...

......1

12313

22321322

2331

ttt

tttt

uuu

XXXY

Koyck lag distribution: geometrically declining weights

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We will see in due course that we can relax both of these constraints.


tttt uYXY 1321

1231211 tttt uYXY

132

23132231

123121321

1

ttttt

tttttt

uuYXX

uuYXXY

...

......1

12313

22321322

2331

ttt

tttt

uuu

XXXY

tttt uYXY 1321

1231211 tttt uYXY

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From the last equation it can be seen that, at time t, with Xt–1, Xt–2, etc already determined, the only influence of X on Y is via Xt. For this reason, again, we describe b2 as the short-run effect.

132

23132231

123121321

1

ttttt

tttttt

uuYXX

uuYXXY


...

......1

12313

22321322

2331

ttt

tttt

uuu

XXXY

2 short-run effect of X on Y

tttt uYXY 1321

1231211 tttt uYXY

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This representation of the model also yields the same long-run effect, as before. The proof is left as an exercise.

132

23132231

123121321

1

ttttt

tttttt

uuYXX

uuYXXY


...

......1

12313

22321322

2331

ttt

tttt

uuu

XXXY

Copyright Christopher Dougherty 2013.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may be

used as a resource for teaching an econometrics course. There is no need to

refer to the author.

The content of this slideshow comes from Section 11.4 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

Additional (free) resources for both students and instructors may be

downloaded from the OUP Online Resource Centre

http://www.oup.com/uk/orc/bin/9780199567089/.

Individuals studying econometrics on their own who feel that they might benefit

from participation in a formal course should consider the London School of

Economics summer school course

EC212 Introduction to Econometrics

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

2013.01.20

http://www.oup.com/uk/orc/bin/9780199567089/

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1 models with a lagged dependent variable a widely-used solution to the problem of including...

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